A trace formula for rigid varieties, and motivic Weil generating series for formal schemes
| dc.creator | Nicaise, Johannes | |
| dc.date | 2007-03-01 | |
| dc.date | 2008-09-26 | |
| dc.date.accessioned | 2026-07-07T10:05:24Z | |
| dc.date.available | 2026-07-07T10:05:24Z | |
| dc.description | We establish a trace formula for rigid varieties $X$ over a complete discretely valued field, which relates the set of unramified points on $X$ to the Galois action on its étale cohomology. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring $R$, and we introduce the Weil generating series of a regular formal $R$-scheme $\mathfrak{X}$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When $\mathfrak{X}$ is the formal completion of a morphism $f$ from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of $f$. When $\mathfrak{X}$ is the formal completion of $f$ at a closed point $x$ of the special fiber $f^{-1}(0)$, we obtain the local motivic zeta function of $f$ at $x$. In the latter case, the generic fiber of $\mathfrak{X}$ is the so-called analytic Milnor fiber of $f$ at $x$; we show that it completely determines the formal germ of $f$ at $x$. | |
| dc.description | To appear in Math. Ann. The original publication is available at http://www.springerlink.com | |
| dc.identifier | https://arxiv.org/abs/math/0703026 | |
| dc.identifier | http://arxiv.org/abs/math/0703026 | |
| dc.identifier | doi:10.1007/s00208-008-0273-9 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/169998 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14G22; 14B05; 32S55 | |
| dc.title | A trace formula for rigid varieties, and motivic Weil generating series for formal schemes | |
| dc.type | text |